2log (problem 3.3.6)

Average Error: 29.0 → 0.1

Time: 11.3s

Precision: 64

Math

\[\log \left(N + 1\right) - \log N\]

↓

\[\begin{array}{l} \mathbf{if}\;N \le 7726.51655081009:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{3}}{N \cdot N}, \frac{1}{N}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]

C

double f(double N) {
        double r1059828 = N;
        double r1059829 = 1.0;
        double r1059830 = r1059828 + r1059829;
        double r1059831 = log(r1059830);
        double r1059832 = log(r1059828);
        double r1059833 = r1059831 - r1059832;
        return r1059833;
}

↓

double f(double N) {
        double r1059834 = N;
        double r1059835 = 7726.51655081009;
        bool r1059836 = r1059834 <= r1059835;
        double r1059837 = 1.0;
        double r1059838 = r1059837 + r1059834;
        double r1059839 = r1059838 / r1059834;
        double r1059840 = log(r1059839);
        double r1059841 = 0.3333333333333333;
        double r1059842 = r1059834 * r1059834;
        double r1059843 = r1059841 / r1059842;
        double r1059844 = r1059837 / r1059834;
        double r1059845 = 0.5;
        double r1059846 = r1059845 / r1059842;
        double r1059847 = r1059844 - r1059846;
        double r1059848 = fma(r1059843, r1059844, r1059847);
        double r1059849 = r1059836 ? r1059840 : r1059848;
        return r1059849;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 7726.51655081009

   1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
   2. Using strategy rm

   3. Applied diff-log 0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]

   if 7726.51655081009 < N

   1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^{2}}}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{3}}{N \cdot N}, \frac{1}{N}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7726.51655081009:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{3}}{N \cdot N}, \frac{1}{N}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))